Damping systems for vibrating members

ABSTRACT

The invention relates to the use of low-density granular fill with a specific gravity less than 1.5, placed in hollow cavities built into turbomachinery blades, vanes, and shafts. The granular fill provides effective damping at lower frequencies than conventional passive damping treatment due to its low bulk compressional sound speed. Selected materials used for the granular fill treatment are chosen for specific turbomachinery needs, including temperature of operation. Rotation speed is also a factor because it induces on the granular material an apparent hydrostatic pressure associated with the centripetal acceleration, and sound speed in granular materials is a function of pressure raised to the power 1/n. Preferred designs for placement of low-density granular material in appendages and shafts are found using an iterative design tool based on the Direct Global Stiffness Matrix method.

RELATED APPLICATION

This is a continuation application of International Application No.PCT/US97/16575 filed on Sep. 17, 1997 and is a continuation-in-partapplication of U.S. Ser. No. 08/731,251 filed on Oct. 11, 1996, now U.S.Pat. No. 5,820,348 and claims priority to U.S. Provisional ApplicationNo. 60/026,234, filed on Sep. 17, 1996, the teachings of theseapplications being incorporated herein by reference in their entirety.

BACKGROUND

Turbomachinery is used in many applications to perform work on orextract work from both gaseous and liquid fluids. Examples of suchmachinery includes gas turbines, axial and centrifugal fans, marine andaviation propellers, fan blades, helicopter blades and tail rotors, windturbines, and steam and hydraulic power turbines. This machinery, bydesign, may contain one or more of a broad class of rotating and fixedappendages including blades, vanes, foils, and impellers depending onthe needs of the particular machine. These appendages are beam-likestructures, often cantilevered, and have natural frequencies ofvibrations, or resonant frequencies, that are excited by mechanicalvibration and fluid flow. In all turbomachinery, power is transmittedvia shafts of one form or another.

Rotating appendages such as gas turbine blades are prone to vibration atcritical speeds, which leads to fatigue and eventually pre-mature, andoften catastrophic, failure of the component. Ensembles of such bladesare components of turbines used as prime movers, such as gas turbines,as well as power generators, such as hydraulic turbines. Vibration ofthe turbine blades is caused by a combination of dynamic effectsincluding imbalance of the rotating system and torsional vibration ofthe power transmission shaft, and fluid dynamic forcing. In certainoperating conditions these phenomenon conspire to excite the naturalmodes of vibration in the turbine blades, and if left unchecked drivethe system to failure. Natural frequencies are defined as thosefrequencies at which an ideal, lossless system will vibrate with zeroinput excitation power. In real systems, which always have a certainamount of intrinsic or added damping, the system will respond at thenatural frequencies and displacement amplitude will grow to the pointthat damping (i.e., conversion of mechanical energy to heat) dominatesor until the part fails.

Fixed appendages such as stator vanes in a gas turbine, as an example,are also subject to dynamic loading, due in part, to the fluid flowdynamics and due, in part, to coupled vibrations from other parts of theturbomachine. Like their rotating counterpart, these fixed appendageshave resonant frequencies, which can be excited by system dynamics.While the fixed appendages do not have the extra load imposed bycentrifugal forces, as do the rotating appendages, excitation of thecomponents at their resonant frequency can still lead to excessivedynamic loads and thence to premature failure.

In all turbomachinery, there are one or more power transmission shaftsto-which the rotating components are attached directly or indirectly. Aswith the other components, the transmission shafts also have resonantfrequencies, which are a function of the shaft geometry, the loadingimposed by the rotating appendages, and the boundary conditions imposedby the locations of the bearings holding the shaft system in place. Atcritical speeds, rotating shafts become dynamically unstable with largelateral amplitudes. Resonance in the shaft, as with the othercomponents, is to be minimized so as to minimize wear on bearings,minimize cyclical fatigue of the shaft, and thus to increase the servicelife and reliability of the equipment.

Of the three vibrating components of turbomachinery, the rotatingappendages are under the most stress and are the most difficult to treatdue, in large part, to the combined effects of mechanical and fluiddynamics, the latter of which is associated with fluid turbulence. Fluideffects on rotating appendages apply as well to the fixed appendages,which are strongly affected by fluid dynamic excitation. Vibrations inshafts are only slightly affected by fluid dynamics, but complicatedmechanical dynamics cause significant loads in some cases with largevibration induced motion.

While mechanical and fluid dynamic loading both result in excitation ofthe cantilever modes of vibration of turbine blades, their causativemechanisms are quite different. Mechanical imbalance of an idealinfinitesimally thin rotor disk, or radial array of turbine blades, onlyproduces a radial force on each blade, and cannot, in principle, excitethe bending cantilever motion that results in blade fatigue. In realsystems, however, two factors contribute to the excitation of bendingmodes in the blades. The first is two-plane rotor imbalance, whichimparts a moment at the base of the blade where it connects to the hub.The second is imperfections in the radial alignment of the turbineblade, which permits purely radial motions of the hub to excite bendingmotions of the blade. Two-plane rotor imbalance tends to excite bendingmotion in a plane parallel to the axis of the power transmission shaftand perpendicular to the plane of the turbine blade disc assembly.Misalignment of the turbine blade tends to convert radial motion of thehub into bending motion in both planes, i.e., that plane parallel to thetransmission shaft and that plane parallel to the turbine blade discassembly.

Further dynamic forcing on the blades results from torsional vibrationsof the transmission shaft. These vibrations are associated with thenatural torsional modes of the shaft assembly and are excited by anytransient event such as changes in speed. Torsional vibration in theshaft couples to bending vibration in the turbine blades with motionprimarily in the plane parallel to the turbine blade disc assembly.

The distinction between the two planes of bending motion is not clearlydefined. Specifically, asymmetry of the blade shape with respect to theplane of rotation causes the two bending directions to be coupled. As aresult, any attempt to minimize bending motion must be effective in bothplanes. The important bending plane, in fact, is that plane which runsthe length of the blade and cuts through its narrowest dimension.

Fluid dynamic loading is a result of vortex shedding at the trailingedge of the (rotating or fixed) blade. Vortex shedding frequencies varyfrom section to section along the length of the blade due to slightvariations in the blade structure and variations of the flow velocityacross the blade. The range of vortex shedding frequencies for any givenblade can thus span a relatively broad bandwidth. If one or more naturalfrequencies of the blade lie within the band of vortex sheddingfrequencies, then the blade will be excited into motion. In shippropellers this phenomenon is known as singing propellers. It has beenfound that blades with relatively straight trailing edges, as is thecase for many turbine blades, are more prone to singing than those withcurved trailing edges. Singing continues to excite the blade untilintrinsic or added damping limits the buildup of displacement amplitude.

Previous treatments for vibration in turbomachinery appendages havefocused on applying damping materials or mechanisms at point locations.The intent is to limit the maximum displacement of the component byconverting the dynamic (kinetic) energy of the appendage into heat,which is innocuous in terms of the performance and service life of themachine. Placing damping treatments at localized points is effective ifthere exist large resonant system dynamics at the chosen point, which isnot always true.

For blades and stator vanes, previous damping treatments have most oftenbeen applied at the base of the appendages, where they attach to therest of the machine, at the tip in the form of a shroud for the blades,and at the inner and outer shroud for vanes. Damping at the base isattractive because the primary modification to the blade or vane is inthe attachment configuration and does not affect the functional shape ofthe foil. In addition, for rotating blades, the extra weight associatedwith the damping treatment is subjected to reduced centrifugal forcesbecause of its proximity to the axis of rotation. Damping at the bladetip by a shroud is effective in reducing the dynamic vibration levels ofcantilevered blades, but comes at the cost of increased weight andcentrifugal forces imposed on the blades and the rotor hub. Intermediatedamping positions have been used in the form of snubbers that arepositioned between the blades at locations part way between the bladeroot and tip. While effective in damping the resonant vibrations,especially if used with a shroud, the snubbers impose extra weight, andin addition, disturb the fluid flow around the appendage, which reducesthe efficiency of the machine.

High temperature gas turbines are especially difficult to treat. In suchsituations complications beyond the high centrifugal force exist.Specifically, the design must deal with heat combined with the fact thatlow order modes of vibration are notoriously difficult to treat usingpassive damping methods. The temperature of operation in gas turbineengines, for example, is in the vicinity of 1300° F., which rendersuseless any conventional viscoelastic polymer or resin. Previousattempts have been made to add ceramic damping layers to the externalsurfaces of turbine blades, but the combination of heat and highcentrifugal force renders the treatment short lived.

Other previous treatments are based on friction devices mounted at theconnections between the blade and the hub. The friction devices rely onthe relative motion between the blade base and the hub. With africtional surface mounted at this location, vibrational energy isextracted from the blade and converted to heat. The shortcoming in thisapproach is that the motion of the blade is low at the junction betweenthe blade and the hub. Effective passive damping is only achieved whentreatments are placed at locations of large displacement.

In another approach, dynamic absorbers have been used to reducevibration levels in many types of devices. A liquid has been placedwithin a chamber of a hollow blade. The liquid oscillates within thechamber, which is sized to produce a resonant frequency approximatelythe same as that of a dominant resonance in the blade. The combinationof the blade resonance and the fluid resonance form, in a simplifiedanalogy, a two degree-of-freedom system in which energy from the blade,which has low intrinsic damping is coupled to energy in the liquid,which through proper selection of viscosity, has high intrinsic damping.The deficiency of this approach is that the dynamic absorber formed bythe liquid oscillator only extracts energy from the blade in arelatively narrow band of frequencies. Since the excitation mechanism isbroadband (a combination of fluid dynamic vortex shedding and mechanicalvibrations with many harmonics) then a narrowband absorber will onlyprovide partial relief. Dynamic absorbers have also been used fordamping shafts.

In still another previous approach, treatment of vibrations associatedwith power transmission shafts and structural acoustics have includedhigh-density granular fill such as sand or lead shot. Broadbandtreatment has been achieved by filling hollow shafts with sand, but theenhanced performance comes at the cost of a substantial weight increasethat is unsuitable for many applications.

SUMMARY OF THE INVENTION

The present invention provides a solution to the shortcomings of theprior art in terms of proper placement of the treatment and broadbandeffectiveness even at low-frequencies. A preferred embodiment of theinvention is directed to the manufacture and use of hollowturbomachinery appendages and shafts that are selectively filled with adamping material. The manufacturing of hollow or chambered members isroutine in the construction of air-cooled vanes and blades for gasturbines and similar techniques can be used for other types of membersor appendages. Hollow transmission shafts are routinely manufactured formany applications.

A low density granular fill material is placed within the hollowportions of the blades, and the shafts of a system having rotatingmembers. Other preferred embodiments include selected distribution offill material for stator vanes, fan blades, various types of ducted andunducted marine propellers, aviation propellers, aircraft airfoils, andothers. The low-density granular fill can be applied without alteringthe shape of the structure being dampened. To reduce vibrations, thepreferred distribution of low-density granular fill in differentappendages and shafts is dependent on the geometry of the material andoperating conditions of the articles. For the purposes of the presentapplication, “low density granular material” is defined as a granularmaterial having a specific gravity of less than 1.5.

Rotor or stator elements, or other members referenced above, are oftenwithin a fluid flowing in laminar or non-laminar form relative to theelement. The effects of fluid flow on the vibrational characteristics ofthe member within the flow can vary greatly, depending upon the natureof the fluid and the forces imparted by the fluid on the member.Vibrations induced by fluid flow are superposed on vibrations caused bymachine imbalance and mechanical vibration. The combination of these twoforcing mechanisms establish the total vibrational characteristics ofthe member.

Structural resonances of an undamped beam can often be reduced 10-20 dBby damping the beam with low density granular fill. These teachings aredescribed in U.S. patent application Ser. No. 08/662,167, filed on Jun.12, 1996, now U.S. Pat. No. 5,775,649, the contents of which isincorporated herein by reference in its entirety. Broadband damping ofinduced vibration is achieved with the application of a low-densitygranular fill within the void of hollow system components. Partialfilling of a hollow structure also results in significant damping if thepositions and component chosen for filling constitute regions of activevibration dynamics. Structures can have a plurality of internal filledchambers positioned to optimize damping. Using this approach, lightweight granular fill is used strategically only where it positivelyreduces system vibration. For many applications, the use of granularmaterials having specific gravities in the range of 0.05 to 0.6 arepreferred. For stationary structures such as stators or moving machinerywhere weight is not critical such as turbomachinery in power plants,granular materials having specific gravities in the range of 0.6 to 1.5can be used.

The choice of fill material for any given application can be important.Specifically, matters of weight, tolerance to high temperature, cost,damping effectiveness, ease of handling, and environmental friendlinessare all important. For high-temperature applications the most importantfeature can be to prevent melting of the granular material within thestructure. Ceramic materials that are both porous (low-density) and havehigh melting points can be readily manufactured, and can serve as atreatment for passive damping of structural vibrations. Such materialsmaintain their integrity even at elevated temperatures.

Ceramics are a broad class of materials usually made from clays, whichare composed chiefly of the aluminosilicate minerals kaolinite, illite,and montmorillonite. The clays are combined with water to make thickpaste slurries which are then formed into shapes and dried at relativelylow temperature, less than about 200° C. The dried forms are then firedat temperatures in the vicinity of 1100° C. to produce the ceramicmaterial, which is resistant to abrasion, heat, and chemical corrosion.Various methods can be employed for producing porous granular ceramicmaterials with specific gravities as low as 0.2 and grain sizes of a fewmillimeters or less. Such material can be used as a low-density granularfill treatment in heat critical applications.

Other materials suitable for high temperature applications include oresor refractory materials that are processed to produce light weight, lowdensity materials such as perlite and vermiculite. These materials canbe fabricated by flash heating of water bearing ore. As the waterevaporates a porous material is produced that remains stable attemperatures above 800° C. and is thus suitable for damping of vibrationat high temperatures.

Added forces due to centrifugal acceleration in rotating appendages isanother difficulty which can be considered and addressed in the designof a damping system. Analytical results have found that the sound speedof an idealized granular material goes as c∝p^(⅙), where c is soundspeed and p is pressure. Results have previously been obtained for leadshot, which is a high-density granular material and not appropriate formany applications. It follows quite closely to the ⅙ power law predictedby theory. In other materials, such as Scotchlite® glass micro-bubbles,the power law goes approximately as p^(¼). With these relations, thesound speed of the granular material can be computed as a function ofrotation speed for the turbine. The low sound speed of the granularmaterial is what enables the attenuation mechanism to become active.With a specification on the rotation speed of the appendage, and othercharacteristics (e.g., size and material), and a knowledge of therelationship between sound speed and pressure, a granular material canbe specified to provide effective damping of the vibrational modes.

The low-density granular fill can also be solid, hollow, spherical inshape, or dendritic in shape. By mixing together two or more types oflow-density granular fill having different characteristics, the soundspeed and overall mass of the fill mixture can be selected.

Not only is it important to choose the proper low-density granular fillmaterial, but the choice of where to place the material is important.Specifically, the material needs to be placed where the system dynamicsare high. Certainly one way to do this is to put the materialeverywhere, but this leads to unnecessary weight in the total structure.To determine placement of the material, software tools, such as theDirect Global Stiffness Matrix (DGSM) method are used to construct aniterative design tool. This method is described in the paper-by J.Robert Fricke and Mark A. Hayner, “Direct Global Stiffness Matrix Methodfor 3-D Truss Dynamics,” ASME 15th Biennial Conference on MechanicalVibration and Noise, Sep. 17-21, 1995 the contents of which isincorporated herein by reference in its entirety. For simple systems,such as beams of uniform cross-section, the system dynamics can becomputed analytically. In such a system, the effect of adding ahigh-loss granular fill subsystem can also be computed analytically oncethe subsystem loss factor, η, is known as a function of frequency. Inmore complicated systems including those with varying beam elementcross-section, varying material properties, and varying pre-stress, theoptimum design cannot be computed analytically except for relativelysimple systems.

Specifically, for turbomachinery members, the cross-section of thestructures vary and the rotating members have varying pre-stress as afunction of radial position due to centrifugal forces. In such caseswhere analytical solutions are not possible, a design tool based on DGSMcan provide information on the optimum placement of granular fillmaterial, and if several materials are available, the proper choice ofmaterial in each location. The design tool is based on an iterativeprocess wherein a cost function is minimized. When the minimum cost isfound, then the system parameters are used to specify the preferreddesign of the damping system. The cost function can be defined indifferent ways for different applications. Factors such as rms velocity,weight of granular material, ease of handling, cost of material, andother criteria can be built into the cost function. Both quantitativeand qualitative measures can be included.

Once the cost function is defined, an iterative search through thedesign parameter space is performed using either deterministicnon-linear search methods, such as steepest decent gradient search, orstochastic non-linear search methods, such as simulated annealing. Theparameters in the design space can be restricted to the placement ofgranular fill, e.g., type and amount as a function of position in thestructure, or can also include design parameters associated with thebase structure, e.g., wall thickness, cross-sectional dimension, ormaterial properties. In the former case, the base structure, say aturbine blade, is taken as a given and only the granular fill dampingtreatment is permitted to vary. In the latter case, the base structureparameters can be varied as well to yield an overall system optimizationbased on the defined cost function.

Direct Global Stiffness Matrix (DGSM) method, which has been formulatedto analyze truss-like structures, is applicable to the case of analyzingturbomachinery appendages and shafts. These structures are slenderobjects, that is, their cross-sectional dimensions are much smaller thantheir length. The DGSM formulation was designed to analyze assemblagesof beam-like structures connected together by “welded” joints, wherewelded simply means the joint is in force balance and displacements arecontinuous. A turbomachinery appendage or shaft, even one of varyingcross-sectional geometry, can be decomposed into an assemblage of localbeam-like structures, each of which has constant cross-sectionalproperties and pre-stress. In this way, any given appendage or shaft canbe modeled by breaking it into a number of beam elements then assemblingthe elements into a whole, which is an approximation of the actualappendage or shaft. Convergence tests show when the discretization ofthe structure is sufficiently fine to permit close approximation to theactual dynamics of the structure being designed. The system can also beused in the analysis of plates, that can be represented, for example, bytwo overlayed orthogonal sets of beams forming a grid, and can similarlybe used in the analysis of other more complex structures.

Other forward models can be used instead of DGSM including statisticalenergy analysis (SEA), finite element analysis (FEA), and boundaryelement analysis (BEA). These methods are known by those skilled in theart and can be used in the design of damping systems as describedherein.

Once the discrete approximation of the appendage or shaft is complete,the iterative design procedure begins. Using the cost function as aguide, design parameters are varied in each beam element according tothe optimization procedure being used (deterministic or stochastic). Ifthere are N design parameters defining the cost function, then theobjective of the optimization procedure is to find the minimum of thecost function in the N-dimensional space. If N is small, say less than10, then deterministic methods work well. If N is large, say greaterthan 100, then stochastic methods work better. If N is between 10 and100, then the preferred optimization method is a function of thecomplexity of the functional relationship of the cost function to the Ndesign parameters. The more complicated the function, the more likelythat deterministic methods are less accurate and that stochastic methodsare used.

The present invention provides effective damping without changing thefunctional form factor (shape) of the machinery. This damping is lightweight with a specific gravity less than 1.5 and does not significantlyincrease requirements for static load capacity of the machinery. Thetreatment can be retrofitted to existing machinery. Low-density granularfill is effective because of its low sound speed, which is a property ofthe selected granular materials.

The present invention can be applied to many types of turbomachineryappendages, including gas turbine blades and vanes, pump impellers, fanblades, marine and aviation propellers, helicopter blades and aircraftwings, wind turbine blades, and steam and hydraulic turbine blades,among others. The damping materials described herein can also be appliedto structures that are mechanically coupled to the above systems orsystems coupled through air, water or other fluids. Thus, the housing orother support structures for turbomachinery, for example, can also bedamped in accordance with the invention.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a schematic drawing of a turbine wheel with blades, hub,shaft, and low-density granular fill.

FIG. 2 is a plot of damping effectiveness of low-density granular fillin steel beams.

FIG. 3 is a plot of sensitivity of low-density granular fill sound speedto changes in pressure.

FIG. 4 is a flow chart for the design method based on DGSM modeling forstructural dynamics.

FIGS. 5A and 5B are schematic drawings showing beam geometry from theside and end views, respectively, for defining the local and globalcoordinate systems.

FIG. 6 is a schematic drawing of a 2-D truss.

FIG. 7 is a schematic drawing of a joint, mass and spring model.

FIG. 8 is a schematic drawing of a 3-D truss.

FIG. 9 is a plot depicting stored energy in truss and total dissipatedpower.

FIG. 10 is a schematic drawing of a propeller blade being modeled asbeam elements.

FIG. 11 is a side view of a rotor and stator blade.

FIG. 12 is a flow chart of a preferred method of installing fill withina structure.

FIGS. 13A and 13B are plots illustrating the damping characteristics ofplates and beams, respectively, in accordance with the invention.

FIG. 14 is a side view of a damped support structure for a damped rotorassembly.

FIG. 15A is a perspective view of a damped housing in accordance withthe invention.

FIG. 15B is a graphical plot illustrating the damping characteristic ofa damped housing for cooling fans.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Referring to a preferred embodiment illustrated in FIG. 1, hollowturbine blades 16 are shown with low-density granular fill 20 for thepassive control of structural vibration. Blades 16 are mounted radiallyon the rotor hub 30, which is connected to the power transmission shaft14. A fluid such as air or water is moving relative to the turbinesystem, such as along axis 12, in either direction parallel to the shaft12. Alternatively, the fluid flow can be in any desired direction, orcan be non-laminar or highly turbulent.

The fractional fill length of the blade (i.e. the volume of the cavityrelative to the volume of the blade) can be anywhere from essentially 0%(i.e., no fill) to essentially 100% (i.e., completely filled). Thus in apreferred embodiment, the cavity volume is substantially filled withdamping material regardless of the ratio of the volume of the cavityrelative to the volume of the blade. The shaft 14 can likewise bepartially or completely filled with low-density granular material 22 toreduce structural vibrations. The blades are closed at the ends tocontain the low-density granular fill 20 in most applications.

In some high temperature applications using metal alloy elements,openings in the elements are employed to provide cooling. Alternatively,ceramic blades can be used for high temperature applications. It ispreferable in most cases to have the fill material remain stationaryrelative to the rotator or stator element except for the low amplitudedisplacements associated with the vibrations being damped.

As an example, for long fan blades in a gas turbine, relative speeds aslow as 10 m/s excite the lowest order vibrational modes due to vortexshedding. For higher velocities, vortex shedding frequencies increaseand excite the higher order modes of the blades. The resonantexcitations must be controlled to prevent catastrophic failures. Thetype of low density granular fill and its distribution in the blade arepreferably chosen during the design stage to minimize resonantvibrations at the operating condition of the engine, which may be sub orsupersonic.

FIG. 2 depicts transfer accelerance measurements for an undamped 5 and adamped 6 steel box beam that is 4 in.×4 in.×8 ft. long. Dampingtreatment includes low-density polyethylene beads. Considerable dampingoccurs in the frequency range of f>200 Hz. Note virtual elimination ofcross-sectional beam resonances above 1000 Hz 7. Only a low frequencyglobal bending mode 8 at about 100 Hz survives the low-density granularfill damping treatment and this mode is reduced by about 20 dB.

FIG. 3 depicts sound speed versus pressure for lead shot (L) 9 andScotchlite™ glass micro-bubbles (B) 10. The best fit power law for thelead shot is c∝p^(1/6.07), which is very nearly the theoretical powerlaw of ⅙ power for spherical particles. The best fit power law for themicro-bubbles is a power law of praised to the exponent 1/4.2 and doesnot fit the theoretical prediction due to the extra compliance of thehollow spheres.

FIG. 4 depicts the iterative method for computing the optimal design fora structure damped using low-density granular fill (LDGF). The systemdesign parameters P_(d), for a system are, for example, the geometricaldimensions and materials, which are fixed for any given design, and areinput into the Direct Global Stiffness Matrix (DGSM) system. The systemcomputes a set of system dynamic responses, q, such as V_(rms), force,and energy density, which are used to compute the cost function F, whichmay also be a function of the free parameters, P_(f). The cost functionis to be minimized over the free parameter space, which are theparameters that can vary for the optimization such as the amount ofprestress imposed on the LDGF and the rotating appendage, density,modulus of elasticity and dimensions. The cost function may also be afunction of design constraints, C, that are limits on the physicalcharacteristics affecting the system, e.g., beam size must always begreater than some specified lower limit. The minimization procedure isachieved by iterating on the values of the free parameters using anadjustment method based on either deterministic non-linear parameterestimation, e.g., steepest decent gradient searches, or stochasticnon-linear parameter estimation, e.g., simulated annealing. Theoptimization procedure is continued until the minimum cost function F,is found.

The low-density granular fill materials employed in the presentinvention may be engineered materials for which one has control of size,shape, material properties and composition. For engineered materials,these properties are carefully controlled and customized to meetperformance criteria for specific applications. These granular materialsare lightweight materials of which polyethylene beads and glassmicro-bubbles are just two examples. These engineered granular materialsare placed inside structural components either at construction time orpoured or inserted at a later time. When the surrounding structure isexcited into vibration, the enclosed granular fill material is excitedas well. The low-density granular fill materials, employed in thepresent invention preferably do not have significant mass relative tothe host structure, nor do they necessarily have high intrinsic dampingas with viscoelastic materials.

The structural damping method and apparatus of the invention offersnumerous advantages over the prior art. It is relatively easy to pourthe granular material into existing structures if retrofitting isrequired for remediation of vibration problems. The lightweight natureof the granular material will not cause significant structural loadingin most cases. Further, if the damping treatment is considered duringthe design stages, there is not a significant increment in static loadbearing requirements. Hence, only modest changes in the structuraldesign are required with respect to that of a design without treatment.

The damping treatment of the invention can be used with closed beams,e.g., hollow sections, as well as with open beams, e.g., I-beams. In thelatter case, only lightweight covering panels are needed to contain thegranular treatment in the vicinity of the structural beam member. Sincethe fill is lightweight, the covering panels do not need to bestructural members, i.e., they can be thin membranes such as plasticsheets or films. Suitable lightweight materials for practice of theinvention include, but are not limited to, a plastic material such aspolyethylene pellets or glass particles such as microspheres ormicro-bubbles. For certain applications polyethylene pellets arepreferred because of the ease of handling and because of slightly betterdamping performance with respect to the glass micro-bubbles. For otherapplications glass micro-bubbles are preferred because they have a bulkspecific gravity in the range of approximately 0.05-0.1 which is atleast an order of magnitude less than the specific gravity of sand (notless than 1.5) and two orders of magnitude less than that of lead shot(7). Because the treatment technique of the invention does not requirehigh density or material viscoelasticity, the choice of materials isquite broad. Such latitude opens he range of material options to includethose that simultaneously offer effective damping as well as otherdesirable traits such as environmental friendliness, ease of handling,cost, etc. Further, since the fill may be an engineered material, theparticle shape, size, and material properties can be tailored to meetspecific performance criteria.

For example, for damping frequencies as low as 200 Hz in tubular steelstructures with cross-sectional dimensions of about 10 cm, low-densitypolyethylene (LDPE) beads which have a roughly spherical shape and aparticle size ranging between about 1-5 mm in diameter are preferredwith about 3 mm in diameter being most preferred. LDPE filled structureswith larger cross-sectional dimensions are damped at even lowerfrequencies. In another example, if light weight is essential, such asin aircraft, glass micro-bubble beads having a spherical shape and aparticle size in the range of about 150-300 microns in diameter arepreferred with about 177 microns in diameter being most preferred fordamping frequencies as low as 300 Hz in tubular aluminum structures withcross-sectional dimensions of about 1 cm. Again, structures with alarger cross-sectional dimensions are damped at lower frequencies.

The damping treatment of the present invention is more effective atlower frequencies than conventional treatments such as constrained layerdamping. Thus, for a given level of effort in damping treatment, thelow-density granular fill permits attenuation of vibrations atfrequencies corresponding to a few bending wavelengths in the structure,which are often troublesome. The frequencies of these low order modesare determined by the structural dimensions and material properties. Inmany cases, these modes occur at a few hundred Hertz, and it has beenshown that LDGF is effective in reducing the resonant dynamics by 10-20dB at such frequencies.

In all cases, the size of the bead treatment particles must be smallenough to be easily accommodated by the structure. The beads should besized such that the voids containing the beads are at least an order ofabout ten times the dimension of the bead diameter. For specialapplications, the beads may be designed for optimum performance. Forexample, if high temperature tolerance is essential as it is for gasturbines, porous ceramic beads having a roughly spherical shape and aparticle size in the range of about 0.1 to 1.0 mm in diameter arepreferred with about 0.5 mm in diameter being most preferred. In anotherexample, if low frequency damping is needed in small structures and heatis not a problem, such as with light aircraft propeller blades, thenlow-density polyethylene (LDPE) beads having a highly irregular,dendritic (needlelike) shape with a nominal diameter of about 2 mm arepreferred as they have a lower sound speed. The lowest frequency forwhich these treatments are effective is dependent on the size of thestructure, larger structures being damped to lower frequencies. It hasbeen shown, however, that resonant frequencies down to about 200 Hz areeffectively damped using the low-density granular fill treatment of thisinvention. Damping to lower frequencies depends on the particulars ofthe structure and treatment design. Damping at frequencies in the rangeof about 100 Hz-500 Hz can now be accomplished using the presentinvention without the additional cost and design constraints imposed byconstrained layer damping.

The theory on which the present invention is based will now be discussedbriefly. The fundamental mechanism that results in damping is that thecompressional wave speed in granular materials, C, is much lower thanthe wave speed in a solid of the same material. With this low wavespeed, the wave length, λ, in the granular matrix is correspondingly lowfor a given frequency f since λ=c/f. With a low sound speed, severaldamping mechanisms, which normally have a small effect, can come intoplay. These mechanisms include the small intrinsic attenuation withinthe solid pieces of granular material, the friction between pieces ofgranular material, and the non-linear hysteresis in the pieces ofgranular material arising from deformation during the wave propagationprocess. Together these mechanisms attenuate the vibrations because manywavelengths occupy a physically small space and the attenuation is astrong function of wavelength, shorter ones being attenuated morereadily. The present invention does not rely on the mass loading effectas known in the prior art, nor does it rely primarily on the highintrinsic attenuation of bulk viscoelastic polymer materials.

The low-density granular material or the “beads” in the presentinvention do not necessarily need to be spherical in shape. What isimportant is that the material be granular so that the bulk wave speedis low. With a low wave or sound speed and hence a small wavelength, anysmall intrinsic attenuation in the material, non-linear hysteresis dueto deformation of the material, or frictional losses between grains ofthe material will provide effective damping of the structural vibration.

Although originally formulated to analyze truss-like structures, aspreviously mentioned, the Direct Global Stiffness Matrix (DGSM) methodcan also be employed to analyze rotating members so that the correctamount and location of low-density granular fill can be determinedrelative to the members. In order to simplify the explanation, thenon-rotating DGSM method is presented first. Once the computation andassembly method is understood, then modification to include rotatingmembers is presented at the end. A discussion of the non-rotating DGSMmethod now follows.

The fundamental building block for the DGSM method is the beam elementin a truss structure. For slender Euler-Bernoulli beams of constantYoung's modulus, E, and cross-section, A, the equations of motion forlongitudinal, torsional, and flexural motion are well documented. Abrief overview of these equations of motion is discussed below with theformulations needed to build the Direct Global Stiffness Matrix systemof equations. The beam geometry shown in FIGS. 5A and 5B is usedthroughout to define the local coordinate system within which the beamdisplacements, rotations, forces, and moments are referenced.

One dimensional compression waves in a beam are calledquasi-longitudinal waves. These waves satisfy the 1-D wave equation

u _(,tt) =c _(L) ² u _(,xx),  (1)

where u is particle displacement along the x-axis and C_(L)={square rootover (E/ρ)} is the longitudinal wave speed. Harmonic solutions toEquation (1) may be written, assuming a temporal dependence of e^(−iωt),as

u=u ⁺ e ^(ik) ^(_(L)) ^(x) +u ⁻ e ^(−ik) ^(_(L)) ^((x−L)),  (2)

where the first term is a right going wave referenced to the left end ofthe beam at x=0 and the second term is a left going wave referenced tothe right end of the beam at X=L and k_(L) is the longitudinalwavenumber. The stress, σ, in the beam is computed from the strain as

 σ=Eu _(,x).  (3)

From this, the total force in the x-direction can be computed as

f _(x) =EAu _(,x),  (4)

where A is the beam cross-section area.

Using Equation (2) and Equation (4) one may now write a 2×2 system ofequations in local coordinates relating the beam amplitudes u⁺ and u⁻ tothe values of beam displacement, u, and force, f_(x) as $\begin{matrix}{{\begin{bmatrix}^{\quad k_{L}x} & ^{{- }\quad {k_{L}{({x - L})}}} \\{\quad {EAk}_{L}^{\quad k_{L}x}} & {{- }\quad {EAk}_{L}^{{- }\quad {k_{L}{({x - L})}}}}\end{bmatrix} \cdot \begin{bmatrix}u^{+} \\u^{-}\end{bmatrix}} = {\begin{bmatrix}u \\f_{x}\end{bmatrix}.}} & (5)\end{matrix}$

In a similar way the torsional waves in a beam may be expressed asmotions satisfying the equation

φ_(x,tt) =c _(T) ²φ_(x,xx),  (6)

where φ_(x) is the rotational displacement of the beam about x-axis, thetorsional wave speed is $\begin{matrix}{{c_{T} = \sqrt{\frac{G}{\rho}}},} & (7)\end{matrix}$

G is the shear modulus of the beam material, and ρ is the materialdensity. Harmonic solutions of Equation (6) may be written in the form

φ_(x)=φ_(x) ⁺ e ^(ik) ^(_(T)) ^(x)+φ_(x) ⁻ e ^(−ik) ^(_(T))^((x−L)).  (8)

The moment or torque on the beam along the x-axis is

m _(x) =GJφ _(x,x),  (9)

where J is the polar moment of inertia.

Using Equation (8) and Equation (9) one may write another 2×2 matrix,this one for the torsional excitation in the beam. Specifically,$\begin{matrix}{{\begin{bmatrix}^{\quad k_{T}x} & ^{{- }\quad {k_{T}{({x - L})}}} \\{\quad {GJk}_{T}^{\quad k_{T}x}} & {{- }\quad {GJk}_{T}^{{- }\quad {k_{T}{({x - L})}}}}\end{bmatrix} \cdot \begin{bmatrix}\varphi_{x}^{+} \\\varphi_{x}^{-}\end{bmatrix}} = {\begin{bmatrix}\varphi_{x} \\m_{x}\end{bmatrix}.}} & (10)\end{matrix}$

For flexural waves about the y-axis, as defined in FIGS. 5A and 5B, thegoverning equation of motion is $\begin{matrix}{{\frac{{EI}_{y}}{\rho \quad A}\quad w_{,{xxx}}} = {- w_{,{tt}^{\prime}}}} & (11)\end{matrix}$

where w is the z component of displacement and I_(y) is the moment ofinertia about the y-axis. The flexural wave speed is dispersive.Specifically, the dependence on frequency is expressed as$\begin{matrix}{C_{B_{y}} = {\sqrt[4]{\frac{I_{y}}{A}}\quad {\sqrt{\omega \quad c_{L}}.}}} & (12)\end{matrix}$

The harmonic solution to Equation (11) may be written as $\begin{matrix}{{w = {{w^{+}^{\quad k_{B_{y}^{x}}}} + {w^{-}^{{- }\quad k_{B_{y}^{({x - L})}}}} + {w_{e}^{+}^{k_{B_{y}^{({x - L})}}}} + {w_{e}^{-}^{- \quad k_{B_{y}^{x}}}}}},} & (13)\end{matrix}$

where the wave amplitudes we correspond to the evanescent flexural wavesin the beam that decay exponentially away from the beam ends. Rotation,bending moments, and shear forces are all related to the wave amplitudew in the z direction as

φ_(y) =−w _(,x),  (14)

m _(y) =−EI _(y) w _(,xx),  (15)

and

f _(z) =−EI _(y) w _(,xxx),  (16)

Using Equation (13) to Equation (16) produces a 4×4 matrix expressingthe flexural excitation on a beam as a function of displacements,rotations, forces, and moments along the beam. In particular, thisbecomes $\begin{matrix}{{\begin{bmatrix}^{\quad k_{B_{y}^{x}}} & ^{k_{B_{y}^{({x - L})}}} & ^{{- }\quad k_{B_{y}^{({x - L})}}} & ^{- \quad k_{B_{y}^{x}}} \\{{- }\quad k_{B_{y}}^{\quad k_{B_{y}^{x}}}} & {{- k_{B_{y}}}^{k_{B_{y}^{({x - L})}}}} & {\quad k_{B_{y}}^{{- }\quad k_{B_{y}^{({x - L})}}}} & {k_{B_{y}}^{- \quad k_{B_{y}^{x}}}} \\{\quad {EI}_{y}k_{b_{y}}^{3}^{\quad k_{B_{y}^{x}}}} & {{- {EI}_{y}}k_{b_{y}}^{3}^{\quad k_{B_{y}^{({x - L})}}}} & {{- }\quad {EI}_{y}k_{b_{y}}^{3}^{\quad {{- }\quad k_{B_{y}^{({x - L})}}}}} & {{EI}_{y}k_{b_{y}}^{3}^{- k_{B_{y}^{x}}}} \\{{EI}_{y}k_{b_{y}}^{2}^{\quad k_{B_{y}^{x}}}} & {{- {EI}_{y}}k_{b_{y}}^{2}^{\quad k_{B_{y}^{({x - L})}}}} & {{EI}_{y}k_{b_{y}}^{2}^{\quad {{- }\quad k_{B_{y}^{({x - L})}}}}} & {{- {EI}_{y}}k_{b_{y}}^{2}^{{- }\quad k_{B_{y}^{x}}}}\end{bmatrix} \cdot \begin{bmatrix}w^{+} \\w_{e}^{+} \\w^{-} \\w_{e}^{-}\end{bmatrix}} = \begin{bmatrix}w \\\varphi_{y} \\f_{z} \\m_{y}\end{bmatrix}} & (17)\end{matrix}$

Flexural bending about the z-axis is only sightly different due to therotation of the local coordinate system and may be expressed in a 4×4matrix as $\begin{matrix}{{\begin{bmatrix}^{\quad k_{B_{z}^{x}}} & ^{k_{B_{z}^{({x - L})}}} & ^{{- }\quad k_{B_{z}^{({x - L})}}} & ^{- \quad k_{B_{z}^{x}}} \\{\quad k_{B_{z}}^{\quad k_{B_{z}^{x}}}} & {k_{B_{z}}^{k_{B_{z}^{({x - L})}}}} & {{- }\quad k_{B_{z}}^{{- }\quad k_{B_{z}^{({x - L})}}}} & {{- k_{B_{z}}}^{- \quad k_{B_{z}^{x}}}} \\{\quad {EI}_{z}k_{b_{z}}^{3}^{\quad k_{B_{z}^{x}}}} & {{- {EI}_{z}}k_{B_{z}}^{3}^{\quad k_{B_{z}^{({x - L})}}}} & {{- }\quad {EI}_{z}k_{B_{z}}^{3}^{\quad {{- }\quad k_{B_{z}^{({x - L})}}}}} & {{EI}_{z}k_{B_{z}}^{3}^{- k_{B_{z}^{x}}}} \\{{EI}_{z}k_{B_{z}}^{2}^{\quad k_{B_{z}^{x}}}} & {{- {EI}_{z}}k_{B_{z}}^{2}^{\quad k_{B_{z}^{({x - L})}}}} & {{EI}_{z}k_{B_{z}}^{2}^{\quad {{- }\quad k_{B_{z}^{({x - L})}}}}} & {{- {EI}_{z}}k_{B_{z}}^{2}^{{- }\quad k_{B_{z}^{x}}}}\end{bmatrix} \cdot \begin{bmatrix}v^{+} \\v_{e}^{+} \\v^{-} \\v_{e}^{-}\end{bmatrix}} = {\begin{bmatrix}v \\\varphi_{z} \\f_{y} \\m_{z}\end{bmatrix}.}} & (18)\end{matrix}$

The next step in developing the Direct Global Stiffness Matrixformulation is to gather equations with displacement and rotation on theright hand side of Equation (5), Equation (10), Equation (17), andEquation (18) into one set and equations with forces and moments intoanother set. Further, to produce a complete set of equations, the waveamplitudes at both ends of each beam need to be evaluated, i.e., at x=0and at x=L. Gathering those equations for displacements and rotationsinto a set yields $\begin{matrix}{{{\begin{bmatrix}D^{0} \\D^{L}\end{bmatrix}\overset{\rightharpoonup}{W}} = {{\begin{bmatrix}{\overset{\rightharpoonup}{U}}^{0} \\{\overset{\rightharpoonup}{U}}^{L}\end{bmatrix}\quad {or}\quad D\overset{\rightharpoonup}{W}} = \overset{\rightharpoonup}{U}}},} & (19)\end{matrix}$

where the wave amplitude vector is defined as $\begin{matrix}{{{\overset{\rightharpoonup}{W}}^{T} = \left\lbrack {u^{+},v^{+},w^{+},\varphi_{x}^{+},w_{e}^{+},v_{e}^{+},u^{-},v^{-},w^{-},\varphi_{x}^{-},w_{e}^{-},v_{e}^{-}} \right\rbrack},} & (20)\end{matrix}$

the displacement vector as $\begin{matrix}{{{\left\lbrack {\overset{\rightharpoonup}{U}}^{0,L} \right\rbrack^{T} = \left\lbrack {u,v,w,\varphi_{x},\varphi_{y},\varphi_{z}} \right\rbrack}}_{{x = 0},L},} & (21)\end{matrix}$

and D^(O,L) are each 6×12 matrices of coefficients, which are complexconstants for a given frequency, beam material, and beam geometry.Similarly, the equations with forces and moments on the right hand sidecan be gathered. These may be written $\begin{matrix}{{{\begin{bmatrix}{- C^{0}} \\C^{L}\end{bmatrix}\overset{\rightharpoonup}{W}} = {{\begin{bmatrix}{\overset{\rightharpoonup}{F}}^{0} \\{\overset{\rightharpoonup}{F}}^{L}\end{bmatrix}\quad {or}\quad C\overset{\rightharpoonup}{W}} = \overset{\rightharpoonup}{F}}},} & (22)\end{matrix}$

where the force vector is defined as $\begin{matrix}{{{\left\lbrack {\overset{\rightharpoonup}{F}}^{0,L} \right\rbrack^{T} = \left\lbrack {f_{u},f_{y},f_{z},m_{x},m_{y},m_{z}} \right\rbrack}}_{{x = 0},L},} & (23)\end{matrix}$

Again, the C^(O,L) are each 6×12 matrices of constant complexcoefficients. The minus sign leading C⁰ is required to properly orientthe force at the x=0 end of the beam in the local beam coordinates.

Now, using the analogy of a spring relating force to displacement asF=Kx, it is desirable to form a local stiffness matrix relating the beamdisplacements to the beam forces . From Equation (19) we have =D⁻¹ andusing this in Equation (22) yields $\begin{matrix}{{{{{CD}^{- 1}\quad \overset{\rightharpoonup}{U}} \equiv {K\overset{\rightharpoonup}{U}}} = \overset{\rightharpoonup}{F}},} & (24)\end{matrix}$

where K is a block stiffness matrix of the form $\begin{matrix}{{K = \begin{bmatrix}K^{00} & K^{0L} \\K^{0L} & K^{LL}\end{bmatrix}},} & (25)\end{matrix}$

which explicitly relates the displacement vectors ^(o) and ^(L) to theforce vectors ⁰ and ^(L). The off-diagonal block elements are equal dueto reciprocity.

Up to this point, the above discussion has been dealing only locallywith a single beam element in its local coordinate system (with thex-axis collinear with the length of the beam). In a real truss, manybeams are assembled into the overall truss structure. Each beam in thefinal configuration is, in general, translated and rotated from theorigin of the global coordinate system. To account for this change fromlocal to global coordinate systems one must construct a transformationmatrix for each beam member. Referring to FIGS. 5A and 5B again, threeglobal coordinates ₁, ₂, ₃ are positioned in the local coordinatesystem. To compute the components of the transformation matrix observethat the unit vector along the local x-axis may be written in terms ofthe global coordinates as $\begin{matrix}{\overset{\rightharpoonup}{x} = {\frac{\left( {{\overset{\rightharpoonup}{g}}_{2} - {\overset{\rightharpoonup}{g}}_{1}} \right)}{\left( {{\overset{\rightharpoonup}{g}}_{2} - {\overset{\rightharpoonup}{g}}_{1}} \right)}.}} & (26)\end{matrix}$

The z-axis unit vector may be derived by using the properties of thecross-product as $\begin{matrix}{\overset{\rightharpoonup}{z} = {\frac{\left( {{\overset{\rightharpoonup}{g}}_{2} - {\overset{\rightharpoonup}{g}}_{1}} \right) \times \left( {{\overset{\rightharpoonup}{g}}_{3} - {\overset{\rightharpoonup}{g}}_{1}} \right)}{{\left( {{\overset{\rightharpoonup}{g}}_{2} - {\overset{\rightharpoonup}{g}}_{1}} \right) \times \left( {{\overset{\rightharpoonup}{g}}_{3} - {\overset{\rightharpoonup}{g}}_{1}} \right)}}.}} & (27)\end{matrix}$

The y-axis follows from the right-hand rule given and . Now, thetransformation matrix from local coordinates to global coordinates maybe written as $\begin{matrix}{{T = \left\lbrack {\overset{\rightharpoonup}{x}\quad \overset{\rightharpoonup}{y}\quad \overset{\rightharpoonup}{z}} \right\rbrack},} & (28)\end{matrix}$

where , , and are column vectors of the local unit vectors expressed interms of the global coordinate system.

Now, a coordinate transformation matrix for the 12×1 vectors and isbuilt as $\begin{matrix}{\underset{’}{T} = {\begin{bmatrix}T & O \\O & T\end{bmatrix}.}} & (29)\end{matrix}$

Applying the coordinate transformation to Equation (19) and Equation(22) and thence to Equation (24) yields the stiffness matrix$\begin{matrix}{{{\left\lbrack {\underset{’}{T}{CD}^{- 1}\quad {\underset{’}{T}}^{T}} \right\rbrack \underset{’}{T}\overset{\rightharpoonup}{U}} = {\underset{’}{T}\overset{\rightharpoonup}{F}}},} & (30)\end{matrix}$

where the matrix in the brackets is the stiffness matrix for anarbitrary beam element expressed in the global coordinate system.Defining new variables for the displacements and forces in globalcoordinates as $\begin{matrix}{{\overset{\rightharpoonup}{\prod\limits_{’}}{= {{\underset{’}{T}\overset{\rightharpoonup}{U}\quad {and}\quad \overset{\rightharpoonup}{\mathcal{F}}} = {\underset{’}{T}\overset{\rightharpoonup}{F}}}}},} & (31)\end{matrix}$

the stiffness formulation for the i^(th) beam is written as$\begin{matrix}{{{\underset{’}{K}\quad {_{j}\quad \underset{’}{\prod\limits^{\rightharpoonup}}}_{i}} = {\overset{\rightharpoonup}{\mathcal{F}}}_{i}},} & (32)\end{matrix}$

where $\begin{matrix}{{\underset{’}{K}}_{i} = {{\underset{’}{T}}_{i}C_{i}D_{i}^{- 1}{{\underset{’}{T}}_{i}^{T}.}}} & (33)\end{matrix}$

There are welded boundary conditions at the joints, but as will later bediscussed, the joints are permitted to be lossy by attaching an externalmass-spring-dashpot system. The welded boundary condition imposes twoconstraints, one on the displacements and one on the forces. Thedisplacement constraint is that all beam ends terminating at a jointmust share the same displacement. This condition may be expressedmathematically as $\begin{matrix}{{{{\quad \underset{’}{\prod\limits^{\rightharpoonup}}}_{i} - {\quad \underset{’}{\prod\limits^{\rightharpoonup}}}_{i + 1}} = {{0\quad {for}\quad i} = 1}},2,\ldots \quad,{M^{j} - 1},} & (34)\end{matrix}$

where M^(j) is the number of beams terminating at joint j. The othercondition that is enforced by a welded joint is one of dynamicequilibrium. Specifically, there must be a net force and moment balanceat the joint. The mathematical statement of this condition may bewritten $\begin{matrix}{{{\sum\limits_{i = 1}^{M^{j}}\quad {\overset{\rightharpoonup}{\mathcal{F}}}_{i}} = {\overset{\rightharpoonup}{\mathcal{F}}}_{ext}^{j}},} & (35)\end{matrix}$

where {right arrow over (ℑ)}^(j) _(ext) is the applied external force tothe j^(th) joint in the truss.

The conditions specified in Equation (34) and Equation (35) are enforcedimplicitly by summing the stiffness from all the beams terminating ateach joint. To illustrate, consider a joint at which two beamsterminate. The force contribution of the first beam is ℑ₁=₁ z,10 ₁,where ₁ is the joint displacement. Similarly, the force contribution ofthe second beam is ℑ₂=₂ ₂. If the joint displacements for all beam endsare equal ₁=₂=, then summing the forces to satisfy Equation (35) isaccomplished equally well by summing the beam stiffnesses, K_(,i). Usingthis approach for satisfying the welded boundary conditions, let usproceed to construct the global stiffness matrix.

The assembly of the global stiffness matrix closely follows the assemblyof finite element static stiffness matrices. All the joints of the trussmust be numbered and the beams terminating at each joint must beidentified. After this is done, the contributions to the globalstiffness matrix due to each beam at each joint must be determined. Itis illustrative to see the global matrix constructed using a concreteexample. Consider the simple 2-D truss shown in FIG. 6. The stiffnessmatrix for each beam relates the displacements and forces of the twojoints at either end. In particular, say for beam B₁, the stiffnessmatrix may be written $\begin{matrix}{{{\underset{’}{K}}_{1} = \begin{bmatrix}{\underset{’}{K}}_{1}^{11} & {\underset{’}{K}}_{1}^{12} \\{\underset{’}{K}}_{1}^{12} & {\underset{’}{K}}_{1}^{22}\end{bmatrix}},} & (36)\end{matrix}$

where the superscripts on the block matrices, ₁ ^(jj), denote the jointnumber G^(j), on either end of the numbered beam, B_(i), designated bythe subscripts on . Using the stiffness matrix for each beam, then, andalso using the summation of stiffness at each joint to implicitlysatisfy the welded joint boundary condition, a global stiffness matrixfor the 2-D truss shown in FIG. 6 may be written as follows:$\begin{matrix}{{\underset{’}{K}}_{1} = {\begin{bmatrix}{{\underset{’}{K}}_{1}^{11} + {\underset{’}{K}}_{3}^{11}} & {\underset{’}{K}}_{1}^{12} & {\underset{’}{K}}_{3}^{13} \\{\underset{’}{K}}_{1}^{21} & {{\underset{’}{K}}_{1}^{22} + {\underset{’}{K}}_{2}^{22}} & {\underset{’}{K}}_{2}^{23} \\{\underset{’}{K}}_{3}^{31} & {\underset{’}{K}}_{2}^{32} & {{\underset{’}{K}}_{2}^{33} + {\underset{’}{K}}_{3}^{33}}\end{bmatrix}.}} & (37)\end{matrix}$

This global stiffness matrix then is used to relate the externallyapplied forces and moments, {right arrow over (ℑ)}_(ext), to the unknownjoint displacements and rotations .

One final step is needed in the construction of the stiffness matrix.Loss in the truss joints can be modeled by considering a simple jointmass and spring model as shown in FIG. 7. Associated with the sixdegrees of freedom in a joint are six masses and springs. To realizeloss in the joint, the spring stiffness in the desired degree of freedomis made complex: this applies to the beam, also. For each joint, thereis a stiffness vector and a mass vector as follows:

k ^(T) =[k _(x) ,k _(y) ,k _(z) ,k _(φ) _(x) ,k _(φ) _(y) ,k ₁₀₀ _(z)],  (38)

and

m ^(T) =[m _(x) ,m _(y) ,m _(z) ,m _(φ) _(x) ,m _(φ) _(y) ,m _(φ) _(z)].  (39)

The extra (complex) stiffness at the joint due to the mass-spring systemmay be written in each degree of freedom as $\begin{matrix}{{{\underset{’}{K}}_{d} = {k_{d} - {\omega^{2}m_{d}}}},} & (40)\end{matrix}$

and the subscript d denotes the particular degree of freedom underconsideration. Assembling these individual stiffnesses into a localstiffness matrix for each joint, one may then add this joint stiffnessto the appropriate diagonal element of the global stiffness matrix likethe one shown in Equation (37). Again, adding the stiffness guaranteesthe welded joint boundary condition of equal displacements and a balanceof forces.

It can be shown that the stiffness matrix , is symmetricpositive-definite, which guarantees that no pivoting is required whensolving the system $\begin{matrix}{{\underset{’}{K}\underset{’}{\prod\limits^{\rightharpoonup}}} = \overset{\rightharpoonup}{\mathcal{F}_{{ext}.}}} & (41)\end{matrix}$

Further, in most cases the stiffness matrix , will be quite sparse,since most beams and joints are not nearest neighbors, as is the casefor the sample 2-D truss configuration in FIG. 6. Thus, fast sparsematrix solvers may be used to efficiently compute the beam displacementsand rotations.

Another point worth considering is that once the system has been solved,in particular, once the inverse stiffness matrix is computed, it iseffectively a harmonic Green's function for the truss. The trussresponse to any given external forcing at the joints may be computed bya matrix multiplication and a weighted superposition of the resultingjoint displacement vectors. Thus, using this formulation, one incurs anup-front cost in inverting the stiffness matrix, but afterward, the costof solving for different forcing arrangements on the truss is trivial.The combination of sparse matrix solvers, guaranteed non-pivotinginversion, and the Green's function nature of the inverted stiffnessmatrix provide an extremely efficient method for studying trussdynamics. For time dependent solutions spectral superposition of thedifferent harmonic solutions may be accomplished through Fouriersynthesis.

The Direct Global Stiffness Matrix method has been used to analyze thedynamics of a 3-D truss, which is shown in FIG. 8.

The structure is made of 6061 T6 aluminum tubing struts with an outsidediameter of 1.27 cm and a wall thickness of 0.165 cm. The materialdensity is 2700 kg/M³ and the Young's modulus is 6.89×10¹⁰ N/M². Thereare 109 struts of three different lengths (48.4 cm, 70.0 cm, and 81.0cm). There are 35 joints, which are machined from 6061 T6 aluminumbar-stock. The outside diameter of the joints is 6.35 cm and the averagemass of the joints is 0.12 kg.

The design described above has no “direct” paths between joints fartherthan one bay away. Two of the DGSM simulation cases described below makeuse of loss factors estimated from measurements on the actual truss.

There are three generic locations that passive damping treatment may beadded to a truss structure: in the beam members, in the joints, and asdynamic absorbers attached to the beams. Five cases dealing with dampingin the beam members and in the joints with varying degrees and placementof damping are considered and are summarized in Table 1.

TABLE 1 FIVE SIMULATION CASES Case # Joint η Beam η Comment 1 0 0.0001Control case 2 0 0.001 Experimentally derived value 3 0.001 0.0001Experimentally derived value 4 0 0.05 Typical achievable value 5 0.050.0001 Typical achievable value

The relative performance of each of these damping treatments isevaluated to provide a guide to the design of an actual full-scale trussstructure. To help in the evaluation, we have developed a performancemetric based on a combination of analytical quantities and engineeringconsiderations. The analytical quantities are energy stored, , and powerlost to damping, p_(d), in the truss. Physically one may think of thesequantities in terms of defining a time-constant for the truss structure.Suppose a continuous signal had been applied to the truss for a longtime with all initial transients having settled out. There will be somekinetic energy stored in the truss dynamics and some potential energystored in the truss elasticity. Naturally at any given time and place,these will be different, but globally over the whole truss the kineticand potential energy will sum to be the total stored energy. At the sametime, the truss is drawing power from the driving source and deliveringthis power either to heat, through absorption mechanisms in the trussitself, or to radiation loss into the air surrounding the truss. Thispower must be exactly balanced by the power delivered by the source fora steady state condition. If the source were suddenly turned off, thestored energy in the truss would bleed out of the system via thedissipation mechanisms. The rate at which the diminution of storedenergy occurs is exactly the energy decay rate of the truss system andmay be represented mathematically as $\begin{matrix}{{Ð_{\omega} = \frac{_{d}}{\mathcal{E}_{s}}},} & (42)\end{matrix}$

which is the narrowband energy decay rate for a given frequency. Ifthere is a great deal of damping relative to the level of stored energy,then the decay rate will be quite high. Similarly, low damping gives asmall decay rate. Before the decay rate is used as part of theperformance metric, however, it must be integrated over the band offrequencies of interest. Thus, the broadband energy decay rate isdefined as $\begin{matrix}{Ð = {\int_{\omega_{1}}^{\omega_{2}}{Ð_{\omega}\quad {\omega}}}} & (43)\end{matrix}$

and implicitly includes the effects of frequency dependent loss.

To compute the dissipated power in the truss, the steady-state inputpower at each of the drive-points is computed. This may be computed asthe time averaged product of the applied force at the drive-point andthe associated joint velocity. The RMS power may then be writtencompactly as $\begin{matrix}{{_{d} = {\frac{\omega}{2}{\sum\limits_{k}{\left\lbrack {\overset{\rightharpoonup}{\mathcal{F}_{{ext},k}}{\overset{\rightharpoonup}{\prod\limits_{’}}}_{k}^{*}} \right\rbrack}_{{{@{drive}} - {{point}\quad k}}}}}},} & (44)\end{matrix}$

where [.] denotes the real part, the term ω derives from the conversionof displacement, , to velocity for harmonic systems, and finally, thesum over k accounts for excitation at many drive-points.

The total potential and kinetic energy stored in the i^(th) beam, , iscomputed using the wave amplitudes, W^(T), which in turn are computedfrom the joint displacements using Equation (19). The potential andkinetic energy stored in each point ^(j) _(s), is computed using lumpedparameter equations for the mass-spring-dashpot systems.

Now, the total stored energy in the truss may be computed as$\begin{matrix}{\mathcal{E}_{s} = {{\sum\limits_{i}\mathcal{E}_{s,i}} + {\sum\limits_{j}{\mathcal{E}_{s}^{j}.}}}} & (45)\end{matrix}$

With the narrowband decay rate , as computed in Equation (42), we thenintegrate over frequency to derive the broadband decay rate.

The energy decay rate, however, is not sufficient to act as aperformance metric on its own. We know a priori that large decay ratesmay be achieved by simply increasing the loss factor η. What is notincluded in the metric is a measure of the engineering cost for a givendamping treatment. Specifically, a weight penalty and a penalty forengineering difficulty can be considered. Although these are notanalytical quantities, they must be included in the performance metricin some way. For purposes of this analysis, the penalties areincorporated as factors on a scale from 1 to 5, with 1 being “good” in anormalized sense and 5 being “bad”. Table 2 shows the penalties proposedfor each of the damping treatments, including the dynamic absorbers,even though they are not included in the analysis here. It is felt thatthe dynamic absorbers are sufficiently well known that by including themin the table, a sense of perspective is provided on the penalty valuesselected for the beam and joint treatments.

TABLE 2 PENALTIES FOR DAMPING TREATMENT Factor Beam Joint Dyn. Abs.Weight, 2 1 5 Eng. Diff.,B 1 5 3

TABLE 3 BROADBAND DECAY RATE FOR FIVE CASES Case # p_(d) (W/N) (J/N)(s⁻¹) 1 3.8 × 10⁶ 8.1 × 10⁷ .05 2 5.3 × 10⁶ 9.8 × 10⁵ 5.4 3 4.3 × 10⁶8.3 × 10⁷ .05 4 4.3 × 10⁶ 1.5 × 10⁴ 287 5 7.4 × 10⁶ 1.3 × 10⁸ .06

From these penalties and the decay rate, an overall performance metricof goodness can be computed as $\begin{matrix}{{\underset{’}{G} = \frac{Ð}{\underset{’}{III}\quad B}},} & (46)\end{matrix}$

where is the final performance metric.

Simulations for the five cases shown in Table 1 were run with varyingamounts of damping in the joints and beams as indicated. For each case,the total stored energy and dissipated power were computed as a functionof frequency. An example of the results for the damped beam with η=0.05,Case 4, is shown in FIG. 9. In this Figure, the total stored energy perunit of input force in the truss is shown as a function of frequency(solid) as is the dissipated power per unit input force (dashed). Fromthese curves, it can be seen that the narrowband energy decay rate isaround 10² at low frequency and 10³ at high frequency. Integrating thesenarrowband results over the frequency range of 100-1000 Hz gives a totalstored energy of 1.5×10⁴ J/N and a dissipated power of 4.3×10⁶ W/N for abroadband energy decay rate of 287s⁻¹. Using this same approach for theother cases, results shown in Table 3 are constructed.

TABLE 4 PERFORMANCE METRIC FOR FIVE CASES Case # B 1 1 1 .05 2 2 1 2.7 31 5 .01 4 2 1 144 5 1 5 .012

These results suggest that effort should be spent in damping treatmentin the beams and not in the joints. The engineering penalty factors arenow applied to arrive at a performance metric for each of the casesunder consideration. The weight and engineering difficulty factors forthe cases and the resulting metric are assumed as shown in Table 4.

For this example, the choice for damping treatment in the truss is inthe beams as shown in Case 4. Damping added to the joints is notnecessary, even for damping loss factors comparable to that applied tothe beams. The physical explanation for this is that the wave energyspends so much more time in the beam that even a small amount of dampingthere will out perform even modest damping applied to the joints.

The above DGSM approach can simulate the dynamics of arbitrary 3-D trussconfigurations and has a number of advantages over existing techniques.These advantages include fast solutions for the global displacement oftruss joints and the ability to compute the wave amplitudes of thedifferent wave types on each beam. Perhaps the most significantadvantage is that the solution to the system of equations is, in fact, aharmonic Green's function, which may be used repeatedly to compute thesolution for the truss motion for an arbitrary combination of appliedforces and moments to the truss joints. This approach is possiblebecause the dynamics are represented as linear and superposition ofelemental forcing solutions is an accurate approximation. Dampingperformance is quantified, thus permitting the designer to make informeddecisions about where and how much damping to use to achieve a givenlevel of performance.

When appendages are rotated about an axis perpendicular to the axis ofthe appendage, then centrifugal forces combined with the distributedmass of the appendage produce tension along the axis of the appendage.This tension increases the flexural rigidity of the appendage and, thus,increases the resonant frequencies. The complication with rotatingappendages, however, is that the tension is not uniform along its lengthas a result of varying mass, centripetal acceleration, and appendagecross-section. The tension induced by the rotation of the appendagegenerally increases toward the hub, which is the principal reason thatreducing weight in rotating parts is so important.

In the context of this invention, another complicating factor inrotating appendages is that the centrifugal forces produce an apparenthydrostatic pressure on any granular material that is contained withinthe appendage. Like the tension imposed on the appendage, the pressureseen by the granular material is not uniform along the length of theappendage. Unlike the case for tension, however, the pressure on thegranular material increases toward the tip of the rotating appendage.The fact that pressure varies with length along the length of theappendage leads to a variable sound speed of the granular material alongthe appendage, and hence a variable loss factor, η. As a result, theloss factor in rotating appendages is a function of rotation speed, Ω,bulk modulus of the granular material, E_(g), bulk density of thegranular material, ρ_(g), and geometry of the material placement, _(g).

The design of a low vibration rotating appendage requires that theappendage have high damping at frequencies that can be excited by eithermechanical dynamics, e.g., vibrations from imbalance and harmonics, orfluid dynamics, e.g., vortex shedding at the Strouhal frequency matchinga structural appendage resonance. A robust design is one that operatesfree of large vibrational dynamics over a broad range of operatingconditions. This requirement must be achieved simultaneously withconsiderations of weight and cost. A methodology to develop such designshas already been discussed in the context of FIG. 4, and a detailedpresentation of the Direct Global Stiffness Matrix (DGSM) method used torepresent a dynamic system of beam-like elements has been presented. Inthe case of rotating members, however, a modification to the DGSM ismade. The flexural bending equations are modified to include tension inthe beam elements.

Recall Equation (11), which is the classical Euler-Bernoulli bendingbeam equation. Tension may be added to the beam to form what is called astiff-string equation. Specifically, $\begin{matrix}{{{\frac{{EI}_{y}}{\rho \quad A}\frac{\partial^{4}w}{\partial x^{4}}} - {\frac{T}{\rho \quad A}\frac{\partial^{2}w}{\partial x^{2}}}} = {- \frac{\partial^{2}w}{\partial x^{2}}}} & (47)\end{matrix}$

where T is the tension on the beam element, and all the other variablesand parameters are as before. If a harmonic solution is assumed, theequation can be rewritten as $\begin{matrix}{{{\frac{\partial^{4}w}{\partial x^{4}} - {\underset{\underset{2\beta^{2}}{}}{\frac{T}{{EI}_{y}}}\frac{\partial^{2}w}{\partial x^{2}}} - {\underset{\underset{\gamma^{4}}{}}{\frac{\omega^{2}\rho \quad A}{{EI}_{y}}}w}} = 0},} & (48)\end{matrix}$

where β and γ are chosen for convenience in calculating the stiff-stringwavenumber, k_(s), as seen next. If a solution of the form w(x)=Ae^(k)^(_(s)) ^(s) is assumed and used in Equation (48) then the dispersionrelation may be written

 k _(s) ⁴−2β² k _(s) ²−γ⁴=0.  (49)

There are four solutions to Equation (49), namely $\begin{matrix}{{k_{s} = {\pm k_{s_{1}}}},{k_{s_{1}}^{2} = {\sqrt{\beta^{4} + \gamma^{4}} + \beta^{2}}}} & (50)\end{matrix}$

and $\begin{matrix}{{k_{s} = {{\pm }\quad k_{s_{2}}}},{k_{s_{2}}^{2} = {\sqrt{\beta^{4} + \gamma^{4}} - {\beta^{2}.}}}} & (51)\end{matrix}$

With these solutions for the stiff-string wavenumber, the harmonicsolution to Equation (47) can be written in a perfect analogy toEquation (13) as $\begin{matrix}{w = {{w^{+}^{\quad k_{s_{2}}x}} + {w^{-}^{{- }\quad {k_{s_{2}}{({x - L})}}}} + {w_{e}^{+}^{k_{s_{1}}{({x - L})}}} + {w_{e}^{-}{^{{- k_{s_{1}}}x}.}}}} & (52)\end{matrix}$

The difference between the solution as written in Equation (52) and thatwritten in Equation (13) is the fact that the wavenumber is now afunction of tension in the beam element. If tension is reduced to zero,then β reduces to zero and the stiff-string wavenumber reduces to$\begin{matrix}{{k_{s}^{2} = {\gamma^{4} = \left\lbrack \frac{\omega^{2}\rho \quad A}{{EI}_{y}} \right\rbrack^{1/2}}},} & (53)\end{matrix}$

which leads directly to the bending wave speed as written in Equation(12) for a beam element with no tension.

In the foregoing, then, a beam bending equation has been derived for usein the DGSM method when the beam elements are in tension, which occurswhen the beam is subjected to centrifugal forces. The solution, aswritten in Equation (52), can be used to assemble the global stiffnessmatrix in the DGSM method. With this equation incorporated into themethod, the representation includes physical characteristics that arespecific to the rotating appendage problem.

As an example of how the modified version of DGSM is used, consider FIG.10 where a physical propeller blade or member 40 rotating about a pivotpoint 50 is represented as an assemblage of four elements 42, 44, 46 and48. The elements are located between radii R₀, R₁, R₂, R₃ and R₄.Propeller blade 40 has three cavities 41, 43 and 45 which are locatedbetween radiuses R₀, R₁, R₂ and R₃. The selection of four elements inFIG. 10 is purely for illustrative purposes, and the actual number ofelements are determined by convergence tests on successively finerapproximations. Certain salient features are illustrated in FIG. 10.Specifically, the beam elements 42, 44, 46 and 48 do not necessarilyhave to be of equal length. This freedom is inherited from thenon-rotating DGSM representation. The centrifugal forces, and hencetensions, arise from the radial dependence of the centripetalacceleration, ω²R, and the spatially distributed mass of the member. Thecalculation of the tension is done automatically for a given geometry ofthe elements and a rotation speed, ω.

Damping of the beam member is achieved by the placement of granularmaterial 47 within individual beam elements 42, 44 and 46 by fillingcavities 41, 43 and 45 with the granular material 47. The damping lossfactor for a given material, rotation speed, and fill geometry isdetermined by interpolation from a lookup table based on experimentalresults for that material. In general the damping loss factor can bewritten as $\begin{matrix}{{\eta = {\eta \left( {\sigma_{p},E_{g},\rho_{g},\overset{\rightharpoonup}{x_{g}}} \right)}},} & (54)\end{matrix}$

where σ_(p) is the apparent hydrostatic pressure imposed on the granularmaterial due to centrifugal forces, E_(g) is the bulk Young's modulus ofthe granular material, ρ_(g) is the bulk density and _(g) represents thegeometric description of the granular material distribution within thestructure. While rotation produces pre-stress in the beam elements,which was described previously, rotation produces an apparenthydrostatic pressure in the granular material, since the material doesnot support shear, and hence, behaves like a fluid. The hydrostaticpressure is lower near the hub, where centrifugal forces are the lowest,and greater near the tip, where centrifugal forces are the greatest.Although cavities 41, 43 and 45 have been depicted in FIG. 10 to beelongate cavities which are positioned along the longitudinal axis ofpropeller blade 40, alternatively, cavities 41, 43 and 45 can bepositioned perpendicular to the longitudinal axis. In addition, cavities41, 43 and 45 can be replaced by a single elongate cavity extendingalong the longitudinal axis of propeller blade 40.

The DGSM program can be used for either fixed (non-rotating) or rotatingappendages in the design cycle illustrated in FIG. 4. In addition, bothfixed and rotating beam elements can be incorporated into a singlerepresentation. An example of this is the assemblage illustrated in FIG.11. The shaft 52, support structure 54 and stator 58 are represented asstatic elements, since centrifugal forces are not dominant if thewhirling of the shaft 52 is ignored. The blades 56 are represented withrotating beam (application of method to plates made up of beams crisscrossing) elements as previously discussed.

Referring to FIG. 12, a preferred method for installing granular fillwithin a structure is shown. In step 60, the location for installing thefill is determined in the manner described above. Then, in step 62, abag (e.g. liner 15 in FIG. 1) or container formed from a polymer, forexample, and having a geometry corresponding to the desired distributionfor the fill is filled with granular fill, evacuated and sealed. The bagcontaining the granular fill is then installed within the structure atthe desired location in step 64. The container can be rigid or flexibledepending upon the needs of a particular application. For systems thatare designed for use with the damping system of the present invention,this modular manufacturing technique can improve safety by minimizingexposure to the granular fill and improve efficiency.

Four materials have been identified for use at elevated temperatures:perlite, vermiculite, ceramic beads, and glass microspheres. The densityrange, approximate sound speed, and approximate maximum operatingtemperature for each of the materials is shown in Table 5 below.

TABLE 5 material properties for high temperature application densityrange appx. appx. max. Material (kg/m³) sound speed (m/s) Temperature (°C.) vermiculite 132 90 1150 perlite  32-200 62 900 glass micro- spheres 70 97 400 ceramic spheres 200-700 appx. 100 1000

Vermiculite is an agricultural, construction, and refractory materialmade by flash heating an ore containing a small percentage of water. Thewater expands and produces a light weight granular material able totolerate high temperatures. Preferred processing of vermiculite involvesgrading particle size to be uniform, and less than 2 mm in diameterpreferably in the range of about 1 mm diameter.

Perlite is a volcanic glass or refractory material that is manufacturedin a manner much the same as vermiculite. A water bearing ore is flashheated to about 870° C. or more, which causes the water to vaporize andproduce a porous, light weight material. Perlite is used in agriculture,construction, and refractory applications and tolerates temperatures upto about 1100° C. Preferred processing for perlite involves milling andgrading the particle size to be less than 2 mm, and preferably less thanabout 1 mm in diameter. The specific gravity of agricultural perlite isabout 0.097 with a large particle size distribution. Milling and siftingthrough a 1.45 mm screen produces a perlite material with a specificgravity of about 0.2. This processing increases the mechanical stabilityof the material in the presence of applies forces.

Porous ceramic granules can be manufactured as an absorbent for liquidsin a variety of applications. The manufacturing process is described inU.S. Pat. No. 5,177,036 entitled “Ceramic material”, the entire contentsof which is incorporated herein by reference. The ceramic beads canrange in size from 1-5 mm and have a bulk specific gravity of 0.2 to0.7. The manufacturing process requires that the beads be fired to atemperature of 1100° C., which makes them stable in elevated temperatureapplications.

Glass microspheres are used because of their light weight and highstrength under static pressure. 3M makes this product under thetradename Scotchlite™. While not a refractory material, per say, themicro-spheres are made of glass and are stable at moderately hightemperatures of several hundred degrees Fahrenheit. The commercialproduct made by 3M is available in several different sizes and grades.The glass micro-spheres can be used in their manufactured form, buthandling of the material is difficult since it becomes airborne soeasily due to its light weight and small particle size. To facilitatehandling, processing of the manufactured material involves making athick paste or slurry of the micro-spheres and a volatile liquid with orwithout adhesives. After the material is in place, modest heat and/or avacuum is applied to accelerate volatilization of the liquid, which isdriven from the mixture leaving the micro-spheres behind. Thetemperature at which the component is treated depends on the volatileliquid. Use of water leaves the micro-spheres in a semi-solid cake, evenwithout any adhesives added. The formation of the micro-sphere cakereduces the occurrence of airborne particulates and reduces the nuisancefactor if the material is intentionally or accidentally released fromthe part in which it is placed. For lower temperature applications analcohol volatile in which trace amounts of shellac were dissolved can beused. When the alcohol is volatilized, the shellac is left behind toenhance surface adhesion between the micro-sphere particles. Othervolatile liquids and mild adhesives can be used for specificapplications.

An experiment using low-density polyethylene (LDPE) beads in contactwith an aluminum plate produced the results shown in the attached FIG.13A. The plate was 10½″×15½″ and 1 mm thick. Three transfer accelerancemeasurements were averaged to produce an average response for the platewith and without low-density granular fill (LDGF) damping. For theundamped case the plate was left bare. For the damped case, a pocketmade of shrink wrap material was formed on one side of the plate andfilled with LDPE beads. The top of the pocket was sealed with tape andthe shrink wrap was heated, causing it to shrink and force the LDPEbeads against the side of the plate. The transfer accelerancemeasurements were repeated and the average was computed. FIG. 13A showsthe results of the two experiments with the light line representing theundamped plate response and the heavy line representing the dampedresponse. The transfer accelerance of the plate is reduced by up to 30dB when the plate is damped.

In FIG. 13B, the drive point accelerance response of a tubular aluminumbeam is shown with and without damping. The beam has an O.D. of 12.7 mmand an I.D. of 9.3 mm and is 0.46 m long. It is suspended using elasticbands with a suspension frequency of about 7 Hz, which simulates a freebeam response. Drive point accelerance was measured at one end of theempty beam and the result is shown as the light line in FIG. 13B. Thebeam was filled with milled and screened perlite and the drive pointaccelerance measurement repeated. The heavy line in FIG. 13B shows theresult. The resonant peaks of the undamped system are reduced by up to15 dB relative to those of the undamped response.

When structures are coupled either directly or indirectly to otherstructures directly excited into vibration by fluid flow or rotatingturbomachinery the coupled structures are similarly excited intovibration. Many such structures can be modeled as beams or plates. Theresults in FIGS. 13A and 13B show the effects of the use of LDGF dampingon such structures. LDGF treatment of the vibrating component results ina dramatic reduction of the vibrating levels. An example of such astructure is shown in FIG. 14 where the rotating shaft 52, or aplurality of rotating elements, of a piece of turbomachinery is coupledto a base structure or housing 57 through a bearing assembly 55 andpossibly through direct excitation from the fluid flow. The basestructure 57 is, in turn, coupled to other structures such as supportbeams 59, which are themselves set into vibration. The vibration levelof any given component is ultimately driven from rotating machinery orfluid flow even though neither force is acting directly on thecomponent. In many cases, however, it is important to reduce thevibration levels of such remote components, and use of LDGF is effectiveto this end.

Structures coupled to fluid and mechanical sources of power are excitedinto vibration, which, in turn, result in the radiation of sound intoany surrounding fluid medium. The medium in many cases is either air orwater. The input power, P_(I), must be exactly balanced by the powerlost to damping in the structure, P_(D), plus the acoustic powerradiated into the surrounding fluid, P_(R). Specifically,

P _(I) =P _(D) +P _(R).  (55)

The power dissipated in the structure, P_(D), may be written as afunction of the structural loss factor, η_(D,) as $\begin{matrix}{{P_{D} = {\omega \quad \eta_{D}m^{''}S{\langle v^{2}\rangle}}},} & (56)\end{matrix}$

where {overscore (ω)} is frequency in radians per second, m″ is the massper unit area, S is the surface and <ν²> is the mean-square velocity ofthe structure. In a similar way, the radiated power, η_(R), may bewritten as $\begin{matrix}{{P_{R} = {\omega \quad \eta_{R}m^{''}S{\langle v^{2}\rangle}}},} & (57)\end{matrix}$

where η_(R) is the radiation loss factor defined as $\begin{matrix}{{\eta_{R} = \frac{\rho \quad c\quad \sigma}{\omega \quad m^{''}}},} & (58)\end{matrix}$

and where σ is the radiation efficiency of the structure (which dependson frequency, structure size, and structure type, e.g., plate, shell, orbeam). Using Equation (56) and Equation (57) in Equation (55) yields$\begin{matrix}{P_{I} = {{\omega \left( {\eta_{D} + \eta_{R}} \right)}m^{''}S{{\langle v^{2}\rangle}.}}} & (59)\end{matrix}$

For a given input power, which is essentially independent of anystructural or radiation damping, an increase in the structural lossfactor, η_(D) must be offset by a reduction in the mean-square velocity,<ν²>. With a reduction in mean-square velocity, the radiated acousticpower P_(R), will reduce by virtue of Equation (57).

Low-density granular fill (LDGF) damping treatments are specificallydesigned to increase the structural loss factor, η_(D). Experiments haveshown that increases by an order of magnitude or more are not uncommonas shown in FIG. 13A and FIG. 13B.

Further insight can be developed by combining Equation (59) and Equation(57) to produce a single equation relating total available input power,P_(I), and the radiated acoustic power, P_(R′); $\begin{matrix}{P_{R} = {\frac{P_{I}}{1 + {\eta_{D}/\eta_{R}}}.}} & (60)\end{matrix}$

This result shows that the ratio of the structural loss factor to theradiation loss factor governs the conversion of input to radiated powerand suggests the investigation of two asymtotic, but practical, cases,i.e., light fluid loading and heavy fluid loading.

For the case of light fluid loading, which is exemplified by airbornesound, the radiation loss factor, η_(R), is always less than thestructural loss factor, η_(D), and usually much less. Thus, for airbornesound the radiated acoustic power can be approximated by $\begin{matrix}{{P_{R} = \frac{P_{I}\eta_{R}}{\eta_{D}}},} & (61)\end{matrix}$

which shows that an increase in structural damping results in areduction of radiated acoustic power.

For the case of heavy fluid loading, as exemplified by water-bornesound, the radiation loss factor, η_(R), is often greater than thestructural loss factor, η_(D), which leads to no simplification ofEquation (60). Nevertheless, even in this case, it is seen that anincrease in structural damping results in a decrease of radiatedacoustic power, though not as dramatic as for the light fluid loadingcase.

With respect to the damping of structure born noise in a housing that ismechanically coupled to a rotating system, measurements of reduction instructure born sound are shown in the ⅓ octave band results of FIG. 15Bof a system like that shown in FIG. 15A. In one embodiment the system isa commercial electronics enclosure 60 excited by three DC brushless fans62, 64, 66 running at about 1750 rpm. The fans produce forcefluctuations on the panel 74 to which they are attached, and thesevibrations are coupled throughout the system and result in forcefluctuations on the base plate 72 of the equipment. The forcefluctuations in the example illustrated in FIG. 15B were measured ateach of three mount points upon which the equipment can sit. The circlesymbols in FIG. 15B show the averaged force autospectrum of theequipment prior to any modification. The equipment was modified so as toincorporate panels containing LDGF material on two sides 70 and the base72. The fans are still forcing an unmodified panel at the front end ofthe enclosure. The solid diamond symbols in FIG. 15B show the reductionin force autospectra when the panels are filled with low-densitypolyethylene (LDPE) beads. The open triangles show the results whenperlite is used. Note-that above 1000 Hz the perlite and LDPE producecomparable levels of reduction, about 15-20 dB. Between 100 Hz and 1000Hz the perlite performs better by 5-10 dB relative to the LDPE. Thelower sound speed of perlite (62 m/s) compared to that of LDPE (95 m/s)reduces the coincidence frequency of the radiating plates and henceincreases loss at lower frequencies.

Any or all of the housing plates or supporting structures such as beamscan be damped as well as the moving elements of the system. In anotherembodiment, the housing 60 can be an acoustic loudspeaker having a basstransducer 64, a mid-range transducer 66 and a high frequency transducer62 mounted on a single panel 74. In one embodiment, the four sidepanels, the top and the bottom can have the damping materials describedherein mounted thereon or used to fill internal panel cavities. Dampingpads 68 having the low density granular fill material therein can beattached to the bottom panel 72 of the speaker to reduce transmission ofvibration from the speaker housing 60 to the floor or other supportsurface on which the housing is situated. Damping pads can also beattached to internal or external surfaces of a structure and which canbe positioned to partially cover the surface. The pads 68 can include alow density granular fill within a plastic bag or film and can beattached with an adhesive such as a single or double sided adhesivetape.

Equivalents

While this invention has been particularly shown and described withreferences to preferred embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the spirit and scope of theinvention as defined by the appended claims.

What is claimed is:
 1. A vibration damped system comprising: a rotatingmember having a cavity; a quantity of low-density granular materialcontained within the cavity of the member, said low-density granularmaterial having a specific gravity less than 0.6 and being positionedwithin the member to damp vibrations of the member.
 2. The member ofclaim 1 wherein the a rotating member is a blade.
 3. The member of claim1 further comprising a container containing the low-density granularmaterial which is inserted within the member.
 4. The member of claim 3in which the container is a bag.
 5. The member of claim 1 in which thelow-density granular material has a sound speed of 90 meters per secondor less.
 6. The member of claim 5 in which the low-density granularmaterial has a diameter less than 1.0 mm.
 7. The member of claim 1 inwhich the low-density granular material is applied to the member withoutaltering the shape of the member.
 8. The member of claim 2 wherein thegranular material is dendritic in shape.
 9. A method of damping avibrating member comprising the steps of: providing the vibratingmember, the member having a cavity; filling the cavity with thelow-density granular material having a sound speed of 90 meters persecond or less providing relative movement between the member and fluidsuch that the low-density granular material damps vibration of themember during rotation of the member.
 10. The method of claim 9 furthercomprising the step of providing a distribution of the low-densitygranular material within the member.
 11. The method of claim 9 furthercomprising the step of filling a container with the low-density granularmaterial and inserting the container in the member.
 12. The method ofclaim 9 in which the low-density granular material comprises arefractory material.
 13. The method of claim 12 in which the low-densitygranular material has a bulk specific gravity in the range ofapproximately 0.05 to 0.6.
 14. The method of claim 9 wherein thelow-density granular material is selected from the group consisting ofglass, ceramic and plastic.
 15. The method of claim 9 further comprisingheating the low-density granular material to an operating temperature ofat least 900° C. is without altering a shape of the member.
 16. Avibration damped rotating system comprising: a rotating member having acavity; a low-density granular material having a specific gravity lessthan 0.6 and being positioned within the cavity to damp vibrations ofthe rotating member.
 17. The vibration damped system of claim 16 furthercomprising a rotating shaft attached to a plurality of rotating members,each member having the low-density granular material within a cavity.18. The vibration damped system of claim 17 wherein the rotating memberscomprise blades of a turbine.
 19. The vibration damped system of claim17 wherein the rotating shaft contains a low-density granular materialthat damps vibrations in the shafts.
 20. The vibration damped system ofclaim 16 wherein the low density granular material comprises glassparticles.
 21. The vibration damped system of claim 16 wherein thelow-density granular material comprises ceramic particles.
 22. Thevibration damped system of claim 16 wherein the low-density granularmaterial comprises plastic particles.
 23. The vibration damped system ofclaim 10 wherein the rotating member comprises a propeller.
 24. Thevibration damped system of claim 16 wherein the low-density granularmaterial has a dendritic shape.
 25. A vibration damped system that issubjected to an external force comprising: a first rotating memberundergoing vibrations from the external force the first member having aquantity of low density granular material to damp the vibrations; and asecond member coupled to the first member, the second member having aquantity of low-density granular material coupled to the second memberto damp vibrations of the second member, the low density granularmaterial having a sound speed of less than 90 meters per second.
 26. Thesystem of claim 25 in which the low-density granular material has aspecific gravity less than 0.6.
 27. The system of claim 25 wherein thegranular material is a refraction material.
 28. The system of claim 25wherein the second member is a housing of the first member.
 29. Themethod of claim 9 further comprising the step of damping mechanicalvibrations caused by the rotation of the member.